Heating a salinity gradient from a vertical sidewall: linear theory

When a body of fluid with a vertical salinity gradient is heated from a single vertical wall, instabilities have sometimes been observed experimentally (Thorpe, Hutt & Soulsby 1969; Chen, Briggs & Wirtz 1971; Tsinober & Tanny 1986). We present a linear stability analysis for this configuration and show that for strong salinity gradients the stability of the fluid to infinitesimal disturbances is governed by a single non-dimensional parameter, \[ Q = \frac{(1-\tau)^6g(\alpha\Delta T)^6}{\nu\kappa_Sl^2(-\beta\overline{S}_z)^5} \] where g is the acceleration due to gravity, α the coefficient of thermal expansion, β the density change due to a unit change in the salinity, ΔT the change of temperature at the wall, $\overline{S}_z$ the vertical salinity gradient, l the horizontal lengthscale (k_Tt)^½, v the kinematic viscosity, (κ_Tt)^½ the diffusivity of heat, k_S the diffusivity of salt and τ the salt/heat diffusivity ratio. This non-dimensional parameter is related to the Rayleigh number, however, it involves two different lengthscales; the penetration depth of the thermal front, l, and the height by which a heated element of fluid would rise in the salinity gradient, $g\alpha \Delta T/(- \beta \overline{S}_z)$. This analysis is valid when the ratio of the vertical lengthscale to the horizontal lengthscale is small. This analysis gives good agreement with the published experimental results.